What should singers do to protect their voice when they have a really bad cold?

we do what everyone else does when they have a bad cold- we take the required medicine, we drink lots of tea, or lemon/honey or whatever warm drink soothes, we try and stay warm and dry, we don't talk or sing too much.
If you absolutely have to perform ( not rehearse, perform) then you need to be very sure that youare fueling your voice from your air supply and not rying to overburden the apparatus in your throat to compensate for the natural lack of resonance. This means extra special attention to good technical singing.
One hard and fast rule: if your voice is not affected, you can generaly sing through a cold. If you have the least bit of vocal discomfort ( itching, scratching, that twitch or pull on the larynx that makes swallowing painful) SHUT UP, and get to the ENT doctor. You don't want an infection to do more damage than it has to.
Best wishes for a speedy recovery

DRINK FLUIDSS!!!! TRY NOT TO COUGH AT ALL SO DRINK A LOT TO CLEAR THE CRUD IN YOUR THROAT! also......nyquill and dayquill lol really really really helps OH and dont try to clear your throat like a lot of people do.....it hurts the chords.

If you make a mixture of warm water and honey (and maybe a drop of lemon juice if you prefer) and drink it reguarly, the drink will soothe your throat and help preserve your voice.

not sing or strain your voice. gargle water & salt mixture

Hot drinks, never strain your voice too much and if you really have to talk, do it in your normal speaking voice. No whispering as it only strains the throat area more and makes it all worse!!! No singing neither!!